How does this site chose its moderators? Whether those moderators reserve the right of banning a user?
There are few blogs from stackoverflow to understand the concept of moderator in SE network :
To know answer for your first question, its better if you go thorugh already asked and answered meta post:
So same happen to our site too, first there was nomination for the Pro Tempore Moderators and then SE staff came up with the decision. This decision was not fully based on nomination only but on various other reason mentioned in above blogs.
There is no election in bet sites, so making moderators is in full hands of SE staff, after graduation we will get democratic election which seems bit far for now.
For your second question about banning:
Yes, moderator do have powers to ban a user if he do what he is not supposed to do like offending, abusing or repeatedly breaking SE policies. But bans are not permanent and can be redeemed and are of variable length. If mod misuse it, you surely can contact SE staff by contact us page.